Releasing the hypothesis that every face of the cube carried
the same equation, in [1]} were presented some
new equations without classification purposes.
A complete classification in this extended setting
was then accomplished by Boll in a series of papers [2]}, [3]},
culminating in his Ph.D. Thesis [4]}.
In these papers the classification of all
the consistent sextupletts of partial difference equations
on the quad graph, i.e., systems of the form (), has been carried out. The only technical assumption used in by Boll is the tetrahedron property. The obtained equations may fall into three disjoint families depending on their bi-quadratics
\(h_{ij}={Q}{y_{k}}{Q}{y_{l}}-Q{Q}{y_{k},y_{l}}, \qquad Q=Q ( y_{1},y_{2},y_{3},y_{4};\alpha _1,\alpha _2 ), \) 
